Graphics Reference
In-Depth Information
10.6.1 Equation of a Straight Line
We start by using a vector b to define the orientation of the line, and a
point
a
in space through which the vector passes. This scenario is shown in
Figure 10.28. Given another point
P
on the line we can define a vector
t
b
between
a
and
P
,where
t
is some scalar. The position vector
p
is given by
p
=
a
+
t
b
(10.60)
from which we can obtain the coordinates of the point
p
:
x
p
=
x
a
+
tx
b
y
p
=
y
a
+
ty
b
z
p
=
z
a
+
tz
b
(10.61)
For example, if
b
=[123]
T
and
a
=(2
,
3
,
4), then by setting
t
=1wecan
identify a second point on the line:
x
p
=2+1=3
y
p
=3+2=5
z
p
=4+3=7
In fact, by using different values of
t
we can slide up and down the line with
ease.
If we already have two points in space
P
1
and
P
2
, such as the vertices of
an edge, we can represent the line equation using the above vector technique:
p
=
p
1
+
t
(
p
2
−
p
1
)
Y
P
t
b
a
p
=
a
+
t
b
b
X
a
Z
Fig. 10.28.
The line equation is based upon the point a and the vector
b
.